Hmm...
Difficult one! Beverley and Bettie I honestly don't know how you would combine them to make a new name...!
Suggestions:Beatrice - has the length of Beverley and includes the 't' of BettieBethany - has similar start of 'Bev' 'Beth' including similar ending of Bettie
Bevie - has the nickname quality of Bettie and is a variant of BeverleyBabette - has the length of Beverley and has the 'ett' sound of Bettie. Bea - as obviously both names start with 'b' sorry running out of ideas

Or you could use variants of the names to make a new combination. For example 'Bettie' is a nickname variant of Elizabeth. Obviously Beverley is a name in her own right. But the name is closely linked to the more modern Waverley as they have a similar sound. Due to this you could combine both Elizabeth and Waverley to make a very pretty more up to date combination that links to your Grandma's. I like the idea of ElizabethWaverley!!